Wavelet Transform Modulus Maxima ridge and its application on Stratigraphic Profiling STUDENT:...

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Wavelet Transform Modulus Maxima ridge and its application on Stratigraphic Profiling

STUDENT: R03521101 CHUN-HSIANG WANG

LECTURER: JIAN-JIUN DING

DATE: 2014/11/27

2Outline

Introduction Wavelet Transformation

Wavelet Zoom Wavelet Transform Modulus Maxima

Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case

Conclusion

3Outline




Conclusion

4Introduction

Why we need Wavelet Transform?

Time Frequency Analysis Wavelet

Short-time Fourier Transform

Wavelet Transform

Time & Frequency Transition & Scaling

Characteristic of Frequency Distinguish local property

5Outline




Conclusion

6Continuous Wavelet Transform

*

,

1( , ) ( ) |

u R s R

t uWf u s f t dt

ss

Wavelet Transform = Dilation + Translation

CWT:

→ Convolution Form

( ) 0t dt

,

1( )u s

t uD t

ss

Dilation

Translation

7CWT-cont.

Basis characteristics

if is a wavelet basis, then

1.

2.

3.

(t)dt 0

2*(t) (t)dt ( ) 1t

2

0

( )d

, ( )u s t

8CWT-cont. Famous Wavelet Basis Type

DerGaussian Mexican Hat

-10 -5 0 5 10-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

t (s)

Differential Gaussian function

-10 -5 0 5 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t (s)

Mexican hat

9CWT-cont. DerGaussain function as example

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t (s

)

u=0, s=0.02u=0.1, s=0.02

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t (s

)

u=0, s=0.02u=0, s=0.04

10Wavelet Zoom Focus on localized signal structures with a

zooming procedure that progressively reduces the scale parameter

11Lipschitz Regularity A function is pointwise Lipschitz at , if

there exists , and a polynomial of degree such that,

The Lipschitz regularity of at or over is the least upper bound of the such that is Lipschitz .

f vK m vp

, ( ) ( )vt f t p t K t v

f v f

[ , ]a b

12Lipschitz Regularity-Example

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

t (s)

Am

ptitu

de

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t (s)

Am

ptit

ud

e

Jump Cusp

Lipschitz alpha =0 Lipschitz alpha =1

13Vanishing moment A wavelet with a fast decay has n vanishing

moments iff there exists with a fast decay such that,

Ψ(t) with n vanishing moments can only “see” a change point with Lipschitz regularity α that is less than n.

( ) 0, [0, )kt t dt k n

14Wavelet Transform Modulus Maxima(WTMM) WTMM = ridge

The dip of equation of this ridge is 0.5 definitely.

1

2( , ) [a,b] , ( , )

1log ( , ) log log( )

2

u s Wf u s As

Wf u s A s

( , 0)jump

15WTMM-cont.

16Outline




Conclusion

17Geotechnical Engineering大地工程 Soil mechanics Rock mechanics, Tunnel Engineering Soil Dynamics, Geotechnical Earthquake

Engineering Engineering Geology, Fault Detecting Foundation Engineering, underground

Excavation ……

凡是地下的工程議題都是大地有關範

疇！

18Cone Penetration Test圓錐貫入試驗 In-situ Test Main Measurement

Cone Resistance,qc

Friction Sleeve,fs

Pore Water Pressure,u2

Target Site investigation

P.K. Robertson, 1990

19Soil Behavior Type(SBT) P.K. Robertson,1998

1. Sensitive, fine grained2. Organic soils (peats)3. Clays (clay to silty clay)4. Silt mixtures (clayey silt to silty clay)5. Sand mixtures (silty sand to sandy

silt)6. Sands (clean sand to silty sand)7. Gravelly sand to sand8. Very stiff sand to clayey sand9. Very stiff, fine grained

0

0

0

'

v

a

a

vttn

vv

sr

P

P

qQ

q

fF

20Ic imply SBT P.K. Robertson, 1998

SBT Description

Ic < 1.31 Gravelly sand to dense sand

1.31< Ic < 2.05 Sands: clean sand to silty sand

2.05< Ic < 2.60 Sand mixtures: silty sand to sandy silt

2.60< Ic < 2.95 Silt mixtures: clayey silt to silty clay

2.95< Ic < 3.60 Clays: silty clay to clay

Ic > 3.60 Organic soil

𝐼 𝑐=√ (3.47−𝑄𝑡𝑛 )2+(1.22+𝐹𝑟 )

2

21Insight of Soil Layers location at which the soil behavior type index

changes abruptly

SBT 6

SBT 3

But in reality……

There will be some noise definitely!

-2 0 2 4 6

0

2

4

6

8

10

12

14

16

18

20

Ic(z)

Dep

th z

(m

)

22Transition Zone Cone can sense a layer boundary up to a distance

of 15 cone diameters ahead and behind.

That will make us more difficult to identify layers !

23Outline




Conclusion

24Simulated Case Demonstration

1 2 3 4

0

2

4

6

8

10

12

14

16

18

20

Ic(z)

De

pth

z (

m)

-1.5 -1 -0.5 0 0.5-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

log(s)lo

gW

I C(u

,s)

A

C

A

C

B

B

25In-situ Case Demonstration-NGES Taxes A&M University

(National Geotechnical Experimentation Site,1993)

26NGES-cont.

27NGES-cont.

28More difficult case Oslo Main airport station

29Oslo in-situ case-cont.

30Outline




Conclusion

31Conclusion WTMM is widely applied to detecting discontinuity,

like jump or cusp, in nowaday engineering. Using a series of scale, or narrowing windows, we

can grab the characteristic of a signal at some one local position.

It’s used to bore one or several holes at a construction site for investigation the stratigraphic property. If we enforce CPT and WTMM in field investigation, it will be more efficient and economical.

32Conclusion-cont. In Taiwan we usually take USCS as main

principle of soil classification but not SBT of CPT. However, it must take lots of time and manpower if we still take USCS.

SBT of CPT has a clear and concise image of civil engineering application, because of the clear distinguishing principle of sand and clay. It will help us to realize a better design in engineering.

33Reference P.K. Robertson, C.E. Wride, Evaluating cyclic liquefaction

potential using the cone penetration test, 1998 P.K. Robertson, Interpretation of cone penetration tests — a

unified approach, 2009 B. S. Chen, P.W. Mayne, Profiling the overconsolidation raito of

clays by Piezocone tests, 1994 Y. Wang, Probabilistic identification of underground soil

stratification using cone penetration tests, 2013 J. Benoît, A. J. Lutenegger, National Geotechnical

Experimentation Sites, 1993 Mallat, A Wavelet Tour of Signal Processing, 2008

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Wavelet Transform Modulus Maxima ridge and its application on Stratigraphic Profiling STUDENT:...

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